U41 Graphs and Properties of Trig Functions: Essential Formulas

The Sine Function: $y = \sin x$

The graph of $y = \sin x$ is a periodic wave. Its domain is all real numbers, $x \in \mathbb{R}$, and its range is $[-1, 1]$. The period is $2\pi$ and the amplitude is $1$.

The Cosine Function: $y = \cos x$

The graph of $y = \cos x$ is also a periodic wave, identical in shape to the sine wave but shifted horizontally. Its domain is $x \in \mathbb{R}$, range is $[-1, 1]$, period is $2\pi$, and amplitude is $1$.

The Tangent Function: $y = \tan x$

The graph of $y = \tan x$ is periodic with vertical asymptotes. Its domain is $x \in \mathbb{R}, x \neq \frac{\pi}{2} + n\pi, n \in \mathbb{Z}$. Its range is all real numbers, $\mathbb{R}$. The period is $\pi$.

General Form: $y = a \sin(bx + c) + d$

For functions of the form $y = a \sin(bx + c) + d$ or $y = a \cos(bx + c) + d$, where $a, b, c, d$ are constants and $b > 0$:

• Amplitude: $|a|$ (the vertical stretch factor).
• Period: $\frac{2\pi}{b}$ (the horizontal stretch/compression factor).
• Phase Shift: $-\frac{c}{b}$ (the horizontal translation).
• Vertical Shift: $d$ (the vertical translation).

Even and Odd Functions

Trigonometric functions exhibit specific symmetry properties:

• Cosine is an even function: $\cos(-\theta) = \cos \theta$.
• Sine and Tangent are odd functions: $\sin(-\theta) = -\sin \theta$, $\tan(-\theta) = -\tan \theta$.

Maximum, Minimum, and Zeros

For the basic sine and cosine functions ($a=1, b=1, c=0, d=0$):

• $y = \sin x$: Maximum value is $1$ at $x = \frac{\pi}{2} + 2n\pi$. Minimum value is $-1$ at $x = \frac{3\pi}{2} + 2n\pi$. Zeros at $x = n\pi$, $n \in \mathbb{Z}$.
• $y = \cos x$: Maximum value is $1$ at $x = 2n\pi$. Minimum value is $-1$ at $x = \pi + 2n\pi$. Zeros at $x = \frac{\pi}{2} + n\pi$, $n \in \mathbb{Z}$.